If I could make up my own argument to Merrill it would be "In what way is this mirror going to save your people. And what do you base this belief on?" She may know that it's a mirror that will take you from place to place, but my question would still be the same. " How does this make the plight of your people better? Explain to me how the benefit outweighs the cost."
I know, Fenris can be just as obstinate and stubbon, but his brand of obstinate doesn't drive me nuts. That's all I'm saying.
I think I have a good answer to that question. She isn't hoping it'll save her people. She's hoping it'll help save what little is left of their heritage. The purpose of the Dalish, the very reason they avoided joining the humans after the sacking of the Dales, is because they are seeking out ancient elven artifacts, the lost language, their ancient culture. Heck, they don't even have their whole religion restored because most of the knowledge is lost.
She is hoping that by recreating an ancient mirror, she will have learned a great deal on ancient elven construction, elven magic, and even elven culture because she could, theoretically, restore a very powerful artifact for her people. Heck, if all the clans had an eluvian, they could contact each other quickly.
Simply think on how fast the elves on Sundermount could have had more halla delivered if they could get in contact with another clan almost instantly.
In her mind, the potential gains far outweigh the risks. It isn't fool-proof but nothing in life is. Heck, Hawke didn't know if he would make a profit going to the deep roads in Act 1, but did anyway because the potential profits certainly beat getting arrested by the templars.